Prime and zero distributions for meromorphic Euler products

TL;DR

This paper investigates the relationship between prime and zero distributions in generalized zeta functions expressed as Euler products, focusing on inequalities linking the function's order and prime multiplicity growth.

Contribution

It introduces a new inequality connecting the meromorphic order of the zeta function with prime distribution multiplicities.

Findings

Established an inequality relating the order of the zeta function to prime multiplicity growth.

Extended understanding of the interplay between zero and prime distributions in meromorphic Euler products.

Provided analytical tools for studying generalized zeta functions with finite order.

Abstract

The aim of the present paper is to study the relations between the prime distribution and the zero distribution for generalized zeta functions which are expressed by Euler products and is analytically continued as meromorphic functions of finite order. In this paper, we give an inequality between the order of the zeta function as a meromorphic function and the growth of the multiplicity in the prime distribution.